number of solutions to a linear equation

gourish · 2014-12-14 03:50:38

if x,y,z belong to N (set of natural numbers)

then the number of solutions to this equation x+y+z= 10 are.... well the answer is given as 36 (from 9C2.... combinations)

can anybody help me in finding the number of solutions.... also help me find the idea to find the number of solutions in any general equation

bobbym · 2014-12-14 04:05:22

To check that answer you must tell me the constraints on x,y and z. Sometimes N starts at 0 and sometimes 1 depending on the book. Admittedly, it usually starts at 1 which would give us 11>x,y,z>0. If so, the answer of 36 can be arrived at in several ways.

You do not know about stars and bars or squares and dashes? You do not know about generating functions? You do not know how to count them up?

gourish · 2014-12-14 04:27:08

yes exactly i am a 12th grade student... pretty good at calculus... but i suffer in combinations... well the constraints on x,y,z are clear the N starts from 1 and not 0 (isn't the set containing 0 as well called 'whole numbers') anyways.... tell me the basic idea i might actually think of getting the solution even without the things u mentioned... (stars and dashes etc...)

yeah i understand a little about generating functions....

bobbym · 2014-12-14 04:31:10

Wikipedia wrote:

There is no universal agreement about whether to include zero in the set of natural numbers. Some authors begin the natural numbers with 0, corresponding to the non-negative integers 0, 1, 2, 3, ..., whereas others start with 1, corresponding to the positive integers 1, 2, 3, ...

but i suffer in combinations...

Combinatorics and Probability are the hardest fields in mathematics. Even people with great reputations make mistakes in them all the time. I live in Vegas, I should know.

gourish · 2014-12-14 04:37:53

well any suggestions on how to solve this current problem....

if u do know any good book or site to practice combinatorics and probability it would be great if you could share them with me..

Last edited by gourish (2014-12-14 04:38:31)

bobbym · 2014-12-14 04:42:03

I will start with the simplest idea. To solve x + y + z = 10 with 11>x,y,z>0 you arrange 10 x's in a row like this,

x x x x x x x x x x

okay?

gourish · 2014-12-14 04:47:42

go it.... but why 10... why not 9 or 11...

bobbym · 2014-12-14 04:49:09

One step at a time.

Take two dashes and place them anywhere between those x's. Here is one arrangement.

x x x _ x x x x _ x x x

follow?

gourish · 2014-12-14 04:52:29

yup... two dashes in the arrangement...

Agnishom · 2014-12-14 04:53:40

bobbym · 2014-12-14 04:54:50

They have split the x's into 3 groups (x x x) (x x x x) (x x x) what do you notice about the number of x's in each group?

gourish · 2014-12-14 04:56:06

@agnishom no i don't but i have seen a formula related to that problem.... i just can't remember it...

bobbym · 2014-12-14 04:58:03

We will get to that. The basics first. What is the answer to my question?

gourish · 2014-12-14 04:58:08

they are greater then 3 but less then 4 in each group of x's @bobbym

Agnishom · 2014-12-14 04:58:17

It is called the Multinomial theorem.

But please pay attention to bobbym's idea now.

bobbym · 2014-12-14 04:58:35

Count and add them...

gourish · 2014-12-14 05:03:15

you mean just add the 3+4+3? or like the total number or arrangements....
if the first thing is what ur asking me to do then it's 12

bobbym · 2014-12-14 05:06:19

Just add them 3 + 4 + 3 = 10. Now here is another one

x x _ x _ x x x x x x x

the two dashes break the 10 x's into (x x)(x)(x x x x x x x)

add them...

gourish · 2014-12-14 05:08:01

sorry my bad... i was counting the dashes as well in my mind... anyways... the answer to this step is again 10... you mean that any such arrangement with two dashes has the sum of such groups equal to 10 right...

bobbym · 2014-12-14 05:09:42

We can think of the three groups of x's as variables and give them names

x = (x x)

y = (x)

z = (x x x x x x x)

gourish · 2014-12-14 05:12:50

yeah... i get it.... so these variables (groups)

gourish · 2014-12-14 05:15:43

the minimum value of any of the variables is 1 and maximum is 8 right.... so if i take x as say 1 then the rest of the two variables are to be selected from the remaining x's right?

bobbym · 2014-12-14 05:15:47

Each one adds up to 10, so wherever we put those dashes between the x's we are actually coming up with solutions to x + y + z = 10 with 11 > x,y,z > 0.

To hammer home this point which is basic to your understanding of what comes later. How would you get one solution to a + b + c + d = 13 with 14 > a,b,c,d > 0?

gourish · 2014-12-14 05:25:13

that's easy just take 13 x's in a line put 3 dashes in between them... it gives me 4 groups of x's... well sum of groups in any such arrangement will give me 13... these groups are my variables a,b,c,d.... right...?

bobbym · 2014-12-14 05:26:33

Yes, show me one example like I have done.

Math Is Fun Forum

#1 2014-12-14 03:50:38

number of solutions to a linear equation

#2 2014-12-14 04:05:22

Re: number of solutions to a linear equation

#3 2014-12-14 04:27:08

Re: number of solutions to a linear equation

#4 2014-12-14 04:31:10

Re: number of solutions to a linear equation

#5 2014-12-14 04:37:53

Re: number of solutions to a linear equation

#6 2014-12-14 04:42:03

Re: number of solutions to a linear equation

#7 2014-12-14 04:47:42

Re: number of solutions to a linear equation

#8 2014-12-14 04:49:09

Re: number of solutions to a linear equation

#9 2014-12-14 04:52:29

Re: number of solutions to a linear equation

#10 2014-12-14 04:53:40

Re: number of solutions to a linear equation

#11 2014-12-14 04:54:50

Re: number of solutions to a linear equation

#12 2014-12-14 04:56:06

Re: number of solutions to a linear equation

#13 2014-12-14 04:58:03

Re: number of solutions to a linear equation

#14 2014-12-14 04:58:08

Re: number of solutions to a linear equation

#15 2014-12-14 04:58:17

Re: number of solutions to a linear equation

#16 2014-12-14 04:58:35

Re: number of solutions to a linear equation

#17 2014-12-14 05:03:15

Re: number of solutions to a linear equation

#18 2014-12-14 05:06:19

Re: number of solutions to a linear equation

#19 2014-12-14 05:08:01

Re: number of solutions to a linear equation

#20 2014-12-14 05:09:42

Re: number of solutions to a linear equation

#21 2014-12-14 05:12:50

Re: number of solutions to a linear equation

#22 2014-12-14 05:15:43

Re: number of solutions to a linear equation

#23 2014-12-14 05:15:47

Re: number of solutions to a linear equation

#24 2014-12-14 05:25:13

Re: number of solutions to a linear equation

#25 2014-12-14 05:26:33

Re: number of solutions to a linear equation

Board footer